Question at position 8 If y' = 6x - 3 and y(2) = 4, then y =6x2 - 3x + 2.3x2 - 3x - 2.6.3x2 - 3x + 2.3x2 - 3x.

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Question at position 8  If y' = 6x - 3 and y(2) = 4, then y =6x2 - 3x + 2.3x2 - 3x - 2.6.3x2 - 3x + 2.3x2 - 3x.

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We start by identifying the given information and what is being asked. The derivative is y' = 6x - 3, and an initial condition is provided: y(2) = 4. We need to find the explicit form of y as a function of x and then compare with the provided answer options.

Step 1: Integrate y' to obtain y in terms of x. Integrating 6x - 3 with respect to x yields y = ∫(6x - 3) dx = 3x^2 - 3x + C, where C is a constant of integration that we will determine using the initial condition.

Step 2: Use the initial condition to solve for C. Substitute x = 2 and y = 4 into y = 3x^2 - 3x + C:
  y(2) = 4 = 3(2)^2 - 3(2) + C = 3(4) - 6 + C = 12 - 6 + C = 6 + C.
To satisfy 4 = 6 + C, we get C = 4 - 6 = -2.
Thus the function is y = 3x^2 - 3x - 2.

Step 3: Compare with the answer options. The provided options are:
- 6x^2 - 3x + 2
- 3x^2 - 3x - 2
- 6
- 3x^2 - 3x + 2
- 3x^2 - 3x

Evaluating each:
- The first option, 6x^2 - 3x + 2, has the wrong coefficient for x^2 (6 instead of 3) and the wrong constant term compared to the derived form, so it is incorrect.
- The second option, 3x^2 - 3x - 2, matches the result of the calculation y = 3x^2 - 3x - 2 exactly, so it is consistent with the integration and initial condition.
- The third option, 6, is a constant function and does not have the x-dependent terms required by y = 3x^2 - 3x - 2, so it is incorrect.
- The fourth option, 3x^2 - 3x + 2, has the correct x^2 and x terms but the constant term is +2 instead of -2, so it is incorrect.
- The fifth option, 3x^2 - 3x, lacks the constant term entirely, so it is also incorrect.

In summary, the correct form derived from the calculus steps is y = 3x^2 - 3x - 2, which corresponds to the second option. The fourth option closely resembles the correct polynomial but with an opposite sign on the constant term, which makes it incorrect given the initial condition.

Question at position 8 If y' = 6x - 3 and y(2) = 4, then y =6x2 - 3x + 2.3x2 - 3x - 2.6.3x2 - 3x + 2.3x2 - 3x.

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